deletions | additions
diff --git a/untitled.tex b/untitled.tex
index f3d0a65..194fc1a 100644
--- a/untitled.tex
+++ b/untitled.tex
...
Then we get
\begin{align*}
&M^>(\omega)=\frac{2 i e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\delta,\nu\mu} \int \frac{d\epsilon}{2\pi}\Biggr\{
\\
& V_{\beta k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{r}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,>}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
&+V_{\beta k}^{*} g_{k}^{h,r}(\epsilon) V_{\alpha k} G^{>}_{\alpha\delta}(\epsilon) V_{\delta q}^* g_{q}^{h,<}(\epsilon) V_{\gamma q} G^{r}_{\gamma\nu}(\omega+\epsilon) \Gamma_{\nu\mu} G^{a}_{\mu\beta}(\omega+\epsilon)(f_e(\omega+\epsilon)+f_h(\omega+\epsilon))
\\
...
Again, the "lesser" term for $M(t,t')$ is obtained by exchanging $1-f$ by $f$ and viceversa.
The last term that we need to compute is $Q>(t,t')+Q<(t,t')= G^{h,>}_{k\gamma}(t,t')G^{<}_{q \beta}(t',t)+ G^{h,<}_{k\gamma}(t,t')G^{>}_{q \beta}(t',t)$. We only calculate $Q^>(t,t')$. For such calculation we employ
\begin{eqnarray}
G^{h,>}_{k\gamma}(\omega+\epsilon) G^{h,>}_{k\gamma}(\epsilon) = \sum_\alpha
[g^{r,h}_{k}(\omega) [g^{r,h}_{k}(\epsilon) V_{\alpha k}
G_{\alpha\gamma}^>(\omega)+g^{>,h}_{k}(\omega) G_{\alpha\gamma}^>(\epsilon)+g^{>,h}_{k}(\epsilon) V_{\alpha k}
G_{\alpha\gamma}^a(\omega)] G_{\alpha\gamma}^a(\epsilon)]
\end{eqnarray}
\begin{eqnarray}
G^{<}_{q
\beta}(\omega) \beta}(\omega+\epsilon) = \sum_\alpha [g^{r}_{q}(\omega+\epsilon) V^*_{\alpha q}
G_{\alpha\beta}^<(\omega+\epsilon)+g^{<}_{q}(\epsilon) G_{\alpha\beta}^<(\omega+\epsilon)+g^{<}_{q}(\omega+\epsilon) V^*_{\alpha q}
G_{\alpha\beta}^a(\epsilon) G_{\alpha\beta}^a(\omega+\epsilon) ]
\end{eqnarray}
Then
\begin{eqnarray}
Q^>(\omega) = \begin{align*}
\frac{ e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} \int
\frac{d\epsilon}{2\pi}\Biggr\{[V^*_{\beta k} g^{r,h}_{k}(\omega+\epsilon) \frac{d\epsilon}{2\pi}\\
&[g^{r,h}_{k}(\epsilon) V_{\alpha k}
G_{\alpha\gamma}^>(\omega+\epsilon)+ V^*_{\beta k} g^{>,h}_{k}(\omega+\epsilon) G_{\alpha\gamma}^>(\epsilon)+g^{>,h}_{k}(\epsilon) V_{\alpha k}
G_{\alpha\gamma}^a(\omega+\epsilon)] G_{\alpha\gamma}^a(\epsilon)] [ V_{\gamma q}
g^{r}_{q}(\epsilon) g^{r}_{q}(\omega+\epsilon) V^*_{\alpha q}
G_{\alpha\beta}^<(\epsilon)+ G_{\alpha\beta}^<(\omega+\epsilon)+ V_{\gamma q} g^{<}_{q}(\epsilon) V^*_{\alpha q}
G_{\alpha\beta}^a(\epsilon) G_{\alpha\beta}^a(\omega+\epsilon) ]
\end{eqnarray} \end{align*}
Then we have
\begin{eqnarray}
Q^>(\omega) \begin{align*}
&Q^>(\omega) = \frac{ e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha} \int \frac{d\epsilon}{2\pi}[\Sigma_{0,\beta\alpha}^r (\omega+\epsilon)G_{\alpha\gamma}^>(\omega+\epsilon)+ \Sigma_{0,\beta\alpha}^> (\omega+\epsilon) G_{\alpha\gamma}^a(\omega+\epsilon)]
[\Sigma^r_{0,\gamma \\
&[\Sigma^r_{0,\gamma \alpha}(\epsilon) G_{\alpha\beta}^<(\epsilon)+ \Sigma^<_{0,\gamma \alpha}(\epsilon)G_{\alpha\beta}^a(\omega)]
\end{eqnarray} \end{align*}
We can split $Q^>(\omega)$ in four contributions
\begin{eqnarray}
Q^{>,1}(\omega) = \frac{ 4e^2}{\hbar^2}\sum_{k\beta,q\gamma,\alpha\mu\nu\tau,\delta\lambda} \int \frac{d\epsilon}{2\pi}[-i\Gamma_{\beta\alpha}]G_{\alpha\delta}^r(\omega+\epsilon)\Gamma_{\delta\lambda} G_{\lambda\gamma}^a(\omega+\epsilon) [-i\Gamma_{\gamma\tau}]G_{\tau\mu}^r(\epsilon)\Gamma_{\mu\nu} G_{\nu\beta}^a(\epsilon)((1-f_h(\omega+\epsilon)+(1-f_e(\omega+\epsilon)))(f_e(\epsilon)+f_h(\epsilon))
